About Angle Converter — Degrees, Radians, Grads & Slope Percent
The angle converter translates a plane angle between the units surveying, engineering, and mathematics actually mix: degrees from drawings and site instructions, radians from every formula that differentiates or integrates, grads (gons) from European survey instruments where a right angle is a clean 100, arcminutes and arcseconds from geodetic coordinates and total-station specs, and revolutions from rotating machinery. Every conversion runs through the degree using NIST SP 811 Appendix B factors — the grad, arcminute, arcsecond, and revolution are exact by definition, and the radian is the exact identity 180/π ≈ 57.29578°.
Slope percent is deliberately NOT in that unit list, because it is not an angle unit: a grade of 8.333% means the rise is 8.333% of the run, so slope % = tan(θ) × 100 — a nonlinear map. Doubling a slope percent does not double the angle (45° is 100%, but 90° would be infinite). The separate slope panel converts angle ↔ slope % ↔ rise:run (the "1 in N" form codes and road signs use), guarded at ±90° where the tangent diverges.
How It Works
- Unit conversion: enter the angle, pick the unit it is in ("from") and the unit you need ("to"). The headline shows the converted value; the table shows the same angle in all six units at once. Every conversion is value × (degrees per from-unit) ÷ (degrees per to-unit), computed in full double precision and displayed to 6 significant digits.
- Slope panel: enter either an angle in degrees or a slope in percent. The panel shows the matching trio — angle, slope percent (tan θ × 100), and the rise:run grade expressed as 1:N where N = 100 ÷ slope %. A 4.7636° ramp is an 8.333% slope, which is the ADA-familiar 1:12.
- The ±90° guard exists because tan diverges there: a vertical face has no slope percent. Angles are accepted up to ±89.99° and slope entries up to the percent that guard can produce (572,957%), so a valid entry on either side always has a partner value on the other.
- Grads make mental survey arithmetic clean — a right angle is 100 grads, so 100 grad = 90° exactly and 1 grad = 0.9° exactly. Arcminutes and arcseconds are the sexagesimal subdivisions (1° = 60′ = 3,600″) used in coordinates and instrument accuracy specs.
Worked Example
A drainage detail calls for a 2% grade and the setting-out plan wants the angle: atan(2/100) = 1.14576° — near the 1.74551% slope of a full 1° but not equal to it, because the map is nonlinear even this close to zero. In the other direction, a math library returns π radians for a straight angle: π rad × (180/π) = 180° exactly, and the same angle is 200 grads and half a revolution. The linear table and the slope panel answer different questions — the first renames an angle, the second computes a tangent.
Degrees to slope percent chart (with radians)
Common angles converted to slope percent (tan θ × 100 — the nonlinear slope relationship) and radians (the linear NIST SP 811 factor), shown to 6 significant digits — exactly what the converter outputs for the same inputs.
| Angle | Slope % | rad |
|---|---|---|
| 1° | 1.74551 | 0.0174533 |
| 2° | 3.49208 | 0.0349066 |
| 5° | 8.74887 | 0.0872665 |
| 10° | 17.6327 | 0.174533 |
| 30° | 57.735 | 0.523599 |
| 45° | 100 | 0.785398 |
Formulas
- Linear conversion through the degree
x = value × f_from ÷ f_to- Degrees per unit (the factor table)
rad 180/π (57.29578) · grad 0.9 · arcmin 1/60 · arcsec 1/3600 · revolution 360- Slope panel (nonlinear — not a unit conversion)
slope % = tan(θ) × 100; θ = atan(slope % ÷ 100); rise:run = 1 : (100 ÷ slope %)
Standards & References
- NIST SP 811, Guide for the Use of the International System of Units, Appendix B.8 — gon (grad) 0.9° (exact); degree = 1.745329×10⁻² rad (= π/180, making 1 rad = 180/π ≈ 57.29578°); revolution = 2π rad (exact); minute 1/60°, second 1/3600° (exact)
- Slope percent = tan θ × 100 (rise over run) — the grade convention of road design, drainage codes, and accessibility standards (e.g. the 1:12 ≈ 8.33% ramp family)
- Displayed to 6 significant digits; conversions computed in full double precision
Frequently Asked Questions
How do I convert degrees to radians?
Multiply degrees by π/180 ≈ 0.0174533. So 30° ≈ 0.5236 rad, 45° ≈ 0.7854 rad, and 180° = π rad exactly. Going the other way, multiply radians by 180/π ≈ 57.29578 — the converter carries the identity in full double precision, so π rad → 180° comes back exact.
Is a 45-degree slope the same as a 100% slope?
Yes — and that is the clearest illustration that slope percent is not proportional to angle. Slope % = tan θ × 100, and tan 45° = 1, so 45° is exactly 100%. But 90° is not 200%: the tangent diverges and the percent becomes infinite. At small angles the relationship looks linear (1° ≈ 1.75%), which is exactly why the converter keeps the slope panel separate from the unit table.
What is a grad (gon) and who uses it?
A right angle divided into 100 parts: 1 grad = 0.9° exactly, so a full circle is 400 grads. European survey instruments and geodetic software (notably in France, Germany, and Scandinavia) use it because decimal right angles make bearing arithmetic clean — perpendiculars are ±100, reversals ±200, with no sexagesimal carrying. 100 grad = 90° exactly in the converter.
How do I convert a slope percent to a rise:run ratio?
Divide 100 by the percent: N = 100 ÷ slope %, and the grade is 1:N (one unit of rise per N units of run). An 8.333% slope is 1:12 — the ADA ramp family; a 2% drainage fall is 1:50; a 100% slope is 1:1. The converter's slope panel shows all three forms at once, and the same N works in any length unit.
Why does the slope panel refuse 90 degrees?
Because slope percent is rise over run, and a vertical face has zero run — tan 90° is undefined and the percent grows without bound as you approach it (89° is already 5,729%, 89.9° is 57,296%). The panel accepts up to ±89.99° and flags anything beyond as having no meaningful slope percent; the linear unit table, which never touches the tangent, converts ±90° and beyond without complaint.